**M=1 kink mode for layer widths comparable to the ion larmor radius **

H.L. Berk, S.M. Mahajan, Y.Z. Zhang

**Abstract**

A kink-tearing eigenmode equation is derived for a slab layer geometry in the limit *m*_{e}/*m*_{i} < β_{i} < *L*^{2}_{n}/* L*^{2}_{s} (*m*_{e}/*m*_{i} = mass ratio, β_{i} ion β value, *L*_{n} = gradient scale length, *L*_{s} = shear length) and when the electron collision frequency is comparable to the eigenfrequency. It is essential for consistency to retain arbitrary ion Larmor radius effects, which are described with the use of the Pade approximation. The asymptotic solution of the inhomogeneous eigenvalue problem is obtained using simple approximations to the eigenfunction. A dispersion relation duplicates previously derived results when a Lorentzian conductivity model is used for electrons, while a new dispersion relation is obtained if electrons are described by kinetic dynamics. The dispersion relation is analytically and numerically investigated. The numerical results are compared to a more complicated and presumably more rigorous asymptotic expression. It is shown that this asymptotic expression requires quite a small value for (*m*_{e}/*m*_{i})β_{i} for accuracy. A fitting formula is found that is more accurate than the asymptotic formula for moderate values of (*m*_{e}/*m*_{i})β_{i}. The dissipationless dispersion relation is discussed and it is shown that local shear at the *q*=1 surface can have a stronger effect than the value of δ*cflx W*_{c} (the magnetohydrodynamics (MHD) energy) on the system's stability properties.

**Action principles for the Vlosov equation: four old, one new **

H. Ye, P.J. Morrison

**Abstract**

Five action principles for the Vlasov--Poisson and Vlasov--Maxwell equations, which differ by the variables incorporated to describe the distribution of particles in phase space, are presented. Three action principles previously known for the Vlasov--Maxwell equations are altered so as to produce the Vlasov--Poisson equation upon variation with respect to only the particle variables, and one action principle previously known for the Vlasov--Poisson equation is altered to produce the Vlasov--Maxwell equations upon variations with respect to particle and field variables independently. Also, a new action principle for both systems, which is called the leaf action, is presented. This new action has the desirable features of using only a single generating function as the dynamical variable for describing the particle distribution, and manifestly preserving invariants of the system known as Casimir invariants. The relationships between the various actions are described, and it is shown that the leaf action is a link between actions written in terms of Lagrangian and Eulerian variables.

**Strong and weak instabilities in a 4-D mapping model of accelerator dynamics **

T. Bountis, S. Tomopaidis

**Abstract**

Periodic solutions of a 4-dimensional (4-D) mapping model of accelerator dynamics are obtained and their stability is studied. It is found that near such unstable periodic orbits of low period, there exist chaotic regions of strong instability in the 4-dimensional space x_{n}, x_{n+1}, y_{n}, y_{n+1}, through which orbits escape very quickly to infinity. On the other hand, near 2-dimensional orbits in the x_{n}, x_{n+1} plane stability conditions are obtained for which the y_{n} oscillations do not grow appreciably even if x_{1}, x_{0} are chosen within the chaotic layer of an associated unstable 2-D orbit. In the latter case, evidence of weak instabilities, or Arnol'd diffusion is found and diffusion coefficients are calculated (≈10^{-11}) and compared with the ones obtained (≈10^{-19}), when x_{1},x_{0} are chosen within a region of oscillatory (quasiperiodic) motion.

**The effect of hot inhomogeneous plasma on geomagnetic micropulsations **

M. Roy, T. Tajima

**Abstract**

Hydromagnetic waves generated in the magnetosphere propagating along the magnetic lines to the ground travel through a medium in which the Alfven velocity increases along the radial direction more rapidly than the variation along the wave vector. The effect of the inhomogeneity perpendicular to the field line on the polarization properties of the magnetic variations of various types of waves is studied, as the field lines corresponding to lower shells are excited by the impulse transmitted from the higher shell field lines. Even though the magnetic variation is predominantly longitudinal at the generation point, after a short distance in the equatorial plane, a transverse component will develop as a result of the plasma inhomogeneity in the direction normal to the magnetic field. This may be connected to the observed phenomenon that micropulsations detected on the ground at different latitudes have different polarization characteristics.

**Fluctuation and thermal energy balance for drift-wave turbulence **

C. Kim, W. Horton, B. Hong

**Abstract**

Energy conservation for the drift-wave system is shown to be separated into the wave-energy power balance equation and an ambient thermal-energy transport equation containing the anomalous transport fluxes produced by the fluctuations. The wave energy equation relates the wave energy density and wave energy flux to the anomalous transport flux and the dissipation of the fluctuations. The thermal balance equation determines the evolution of the temperature profiles from the divergence of the anomalous heat flux, the collisional heating and cooling mechanisms and the toroidal pumping effect.

**Temperature-gradient instability induced by conducting end walls **

H.L. Berk, D.D. Ryutov, Yu. A. Tsidulko

**Abstract**

A new rapidly growing electron-temperature-gradient instability is found for a plasma in contact with a conducting wall. The linear instability analysis is presented and speculations are given for its nonlinear consequences. This instability illustrates that conducting walls can produce effects that are detrimental to plasma confinement. This mode is of importance in open-ended systems such as mirror machines and relevant to the edge of tokamaks where field lines are open and are connected to limiters or divertors and astrophysical plasmas like the ones of the flux tubes in a solar atmosphere, with the footpoints on the photospheric level.

**Ion temperature gradient driven turbulence in the weak density gradient limit **

S. Hamaguchi, W. Horton

**Abstract**

The anomalous heat transport arising from the ion temperature gradient driven mode or η_{i}-mode turbulence is extended to the range of the weak density gradient limit (η_{i}=*L*_{n}/*L*_{T} → ∞), which is appropriate for H-mode dishcarges. It is shown that the anomalous ion heat conductivity χ_{i} with *L*_{ n} → ∞ scales as χ_{i} = *g*(ρ_{s}/*L*_{T}) (*cT*_{i}/*eB*) ^{(-βσ)} with σ=(*T*_{e}/*T*_{i}) (* L*_{T}/*L*_{s}), β ≈ 4, and *g* ≈ 1. This χ_{i} scaling is the natural extension for high η_{i} of the scaling of χ_{i} for *K*=(*T*_{i}/*T*_{e}) (1+η_{i}) < 4 obtained (Phys. Fluids B 2, 1833 (1990)) from analytical and numerical studies.

**Bent crystal septa for beam extraction from accelerators **

B.S. Newberger

**Abstract**

In this brief report, we summarize our discussion of the application of the phenomenon of the channeling of charged particles in curved single crystals which has been subject of several recent talks.

**A nonlinear bounce kinetic equation for trapped electrons **

F.Y. Gang, P.H. Diamond

**Abstract**

A nonlinear bounce-kinetic equation that is suitable for studies of nonlinear trapped-electron dynamics in the presence of short wavelength electrostatic fluctuations is systematically derived. The equation is used to analyze the nonlinear response of trapped electrons to the short wavelength collisionless trapped-electron mode in a sheared magnetic field. It is shown that nonlinear trapped-electron--wave interaction (trapped-electron Compton scattering) transfers wave energy from long to short wavelengths. However, the transfer rate is significantly restricted by magnetic shear, which limits the interaction to a narrow region near the rational surface that is typically smaller than the linear mode width. Therefore, it is expected that, in a sheared magnetic field, ion Compton scattering rather than nonlinear trapped-electron--wave interaction will be the dominant nonlinear process in short wavelength trapped-electron mode turbulence.

**Extremal bounds on drift wave growth rates and transport **

T.K. Fowler, P.J. Morrison

**Abstract**

A variational technique is used to obtain bounds on the growth constant {gamma} versus wave number {kappa} for plasma drift waves. We find, for T_{i} = T_{e}, γ<√2ω_{*}(1 + 3/√ 2η) in usual notation. This agrees closely with dispersion---relation results that have had good success in explaining global confinement times in tokamaks based on transport coefficients of the form (γ/κ^{2}). The present method is easier to calculate and results are of such general nature as to give greater assurance that nothing has been missed. The method is based on the time behavior of a free energy function that is chosen to be a constant of motion for an idealized Maxwellian plasma without currents, and almost constant for small departures from this ideal state. The underlying premise associating the variational technique with drift waves remains conjectural.

**Resonant MHD modes with toroidal coupling, Part I. Tearing modes **

J.W. Connor, R.J. Hastie, J.B. Taylor

**Abstract**

In a cylindrical plasma, tearing modes can be calculated by asymptotic matching of ideal mhd solutions across a critical layer. This requires a quantity Δ' which represents the discontinuity' in the ideal solution across the layer. In a torus, poloidal harmonics are coupled and there are many critical surfaces for each toroidal mode number, and correspondingly many discontinuities Δ'_{m}. The ideal mhd solutions do not determine the Δ'_{m} but only a relation between them--described by an E-matrix. We discuss the calculation of the E-matrix for a large aspect-ratio tokamak. In a weak-coupling approximation it is tri-diagonal and can be computed from integrals over the uncoupled eigenfunctions or from simple basis-functions comprising triplets of coupled poloidal harmonics. This weak coupling approximation fails if Δ'_{m} is already small for an uncoupled harmonic. An alternative strong-coupling approximation is developed for this case.

**Nonlinear interaction of photons and phonons in electron-positron plasmas **

T. Tajima, T. Taniuti

**Abstract**

Nonlinear interaction of electromagnetic waves and acoustic modes in an electron-positron plasma is investigated. The plasma of electrons and positrons is quite plastic so that the imposition of electromagnetic (em) waves causes depression of the plasma and other structural imprints on it through either the nonresonant or resonant interaction. Our theory shows that the nonresonant interaction can lead to the coalescence of photons and collapse of plasma cavity in higher (≥2) dimensions. The resonant interaction, in which the group velocity of em waves is equal to the phase velocity of acoustic waves, is analyzed and a set of basic equations of the system is derived via the reductive perturbation theory. We find new solutions of solitary types: bright solitons, kink solitons, and dark solitons as the solutions to these equations. An implication of the present theory on astrophysical plasma settings is suggested, including the cosmological relativistically hot electron-positron plasma.

**Wave-particle power transfer in a steady-state driven system **

H.L. Berk, B.N. Breizman, S. Hamaguchi

**Abstract**

The general expression of the power transfer from a high-energy ion beam to a background electrostatic plasma wave is obtained for arbitrary wave amplitude. It is verified that phase space gradients produced by a finite amplitude wave enhance the power transfer significantly.

**Stability of the gas dynamic trap **

H.L. Berk, G.V. Stupakov

**Abstract**

The description of stability of the gas dynamic trap is shown to be extremely sensitive to the nature of the boundary conditions. Two model boundary conditions in a moderately long mean free path limit are considered: insulating boundary conditions and conducting boundary conditions. The former boundary condition reproduces the MHD results of previous studies, where the outflow of ions contribute to the MHD stabilizing properties of the system. However, with conducting boundary conditions it is shown that the outflowing ions do not contribute to the system's stability. In this case, which is likely to be physically relevant, the MHD stabilizing term only comes from the electron pressure in the expansion region, and the gas dynamic trap would not be as stable as previously envisioned.

**Particle simulation model of the Lorentz collision operator in guiding-center plasmas **

J.H. Han, J.N. Leboeuf

**Abstract**

A simulation model of the Lorentz collision operator has been developed for guiding-center electron plasmas. In this model both the energy and the magnetic moment of the magnetized electrons are conserved. We implemented a two-and-one-half-dimensional (21/2-D) guiding-center particle code with this model to study low-frequency electrostatic resistive interchange modes in sheared slab geometry. The growth rates and linear eigenmode widths obtained from the particle simulation and from linear kinetic theory agree very well.

**Direct conversion of muon catalyzed fusion energy **

T. Tajima, S. Eliezer, R.M. Kulsrud

**Abstract**

In this paper a method of direct conversion of muon catalyzed fusion (MCF) energy is proposed in order to reduce the cost of muon production. This MCF concept is based on a pellet composed of many thin solid deuterium-tritium (DT) rods encircled by a metallic circuit immersed in a magnetic field. The direct energy conversion is the result of the heating of the pellet by beam injection and fusion alphas. The expanding DT rods causes the change of magnetic flux linked by the circuit. Our calculation shows that the direct conversion method reduces the cost of one muon by a factor of approximately 2.5 over the previous methods. The present method is compatible with a reactor using the pellet concept, where the muon sticking is reduced by the ion cyclotron resonance heating and the confinement of the exploding pellet is handled by magnetic fields and the coronal plasma.

**The energy-momentum tensor for the linearized Maxwell-Vlasov and kinetic guiding center theories **

D. Pfirsch, P.J. Morrison

**Abstract**

A modified Hamilton--Jacobi formalism is introduced as a tool to obtain the energy-momentum and angular-momentum tensors for any kind of nonlinear or linearized Maxwell-collisionless kinetic theories. The emphasis is on linearized theories, for which these tensors are derived for the first time. The kinetic theories treated---which need not be the same for all particle species in a plasma---are the Vlasov and kinetic guiding center theories. The Hamiltonian for the guiding center motion is taken in the form resulting from Dirac`s constraint theory for nonstandard Lagrangian systems. As an example of the Maxwell-kinetic guiding center theory, the second-order energy for a perturbed homogeneous magnetized plasma is calculated with initially vanishing field perturbations. The expression obtained is compared with the corresponding one of Maxwell--Vlasov theory.

**The diagnostic possibilities of heated electrons - A simplified view **

W. Thompson

**Abstract**

The possibility of using a population of energetic electrons produced in a small well-defined region as a tracer for magnetic field lines and a diagnostic for anomalous transport is examined in its simplest form. The population is produced by electron cyclotron resonance heating, and the needed power input estimated; its is detected by the emitted bremsstrahlung and the flux is calculated. If this is inadequate to permit the tracing of field lines, information can be gained by examining the spectrum, since electrons passing twice through the local heating region will gain double energy, hence near rational surfaces may be identified. The method appears promising, but open questions are indicated.

**Instabilities and vortex dynamics in shear flow of magnetized plasmas **

T. Tajima, W. Horton, P. Morrison, J. Schutkeker, T. Kamimura, K, Mima, Y. Abe

**Abstract**

Gradient-driven instabilities and the subsequent nonlinear evolution of generated vortices in sheared ExB flows are investigated for magnetized plasmas with and without gravity (magnetic curvature) and magnetic shear by using theory and implicit particle simulations. In the linear eigenmode analysis, the instabilities considered are the Kelvin--Helmholtz (K--H) instability and the resistive interchange instability. The presence of the shear flow can stabilize these instabilities. The dynamics of the K--H instability and the vortex dynamics can be uniformly described by the initial flow pattern with a vorticity localization parameter ε. The observed growth of the K--H modes is exponential in time for linearly unstable modes, secular for the marginal mode, and absent until driven nonlinearly for linearly stable modes. The distance between two vortex centers experiences rapid merging while the angle θ between the axis of the vortices and the external shear flow increases. These vortices proceed toward their overall coalescence, while shedding small-scale vortices and waves. The main features of vortex dynamics, the nonlinear coalescence and the tilt or the rotational instabilities of vortices, are shown to be given by using a low-dimension Hamiltonian representation for interacting vortex cores in the shear flow.

**Transition from resistive-g to η _{i} driven turbulence in stellarator systems **

B.G. Hong, W. Horton, S. Hamaguchi, M. Takatani, M. Yagi, H. Sugama

**Abstract**

By an electromagnetic incompressible two fluid model describing both ion temperature gradient drift modes (eta(sub i) modes) and resistive interchange modes (g modes), a new type of η_{i} mode is studied in cylindrical geometry including magnetic shear and an averaged curvature of Heliotron/Torsatron. This η_{i} mode is destabilized by the coupling to the unstable g mode. Finite plasma pressure beta increases the growth rate of this mode and the radial mode width also increases with plasma pressure β indicating large anomalous transport in the Heliotron/Torsatron configuration. The transport from η_{i} mode exceeds that from resistive g when the mean-free-path exceeds the machine circumference. For plasma β above two to three times the Suydam limit the m = 1/n = 1 growth rate increases from the η_{i} mode value to the MHD value.

**Nonlinear drift waves and transport in magnetized plasma **

W. Horton

**Abstract**

Recent developments in the physics of nonlinear drift waves are presented with special emphasis on the formulation and solutions of reduced hydrodynamic drift wave models. The coherent vortex solutions, the renormalized turbulence theory and the chaotic three-wave interactions are analyzed. The intrinsic stochasticity of the particle motions is analyzed for several types of drift wave systems, and the resulting diffusivities are applied to several tokamaks.

**Enhanced pinch effect due to electrostatic potential **

K.C. Shaing, R.D. Hazeltine

**Abstract**

An inward pinch appears necessary to explain experimental results in tokamaks. Neither the neoclassical pinch effect, which is too small, nor the off-diagonal quasilinear term, which is usually outward in the trapped particle regime, can account for the observations. A mechanism for an enhanced inward pinch is proposed, based on results for an asymmetric magnetic field bump. Because turbulent fluctuations also break toroidal symmetry, an enhanced inward pinch driven by the fluctuations and the Ohmic inductive field, E is expected. To demonstrate this effect, an inward particle flux is calculated for a model tokamak configuration that has an electrostatic potential bump Φ_{0} at toroidal angle ζ = 2π. For the parameter regime r/R < eΦ_{0}/T_{ e} < 1, the flux is found to be Γ = -4.47K (q) (r/R) L(Φ_{0})(υ_{ te}/Rνe)^{1/2}cN E/B, where r(R) is minor (major) radius, B is the magnetic field strength, υ_{te} is the electron thermal speed, ν_{e} is the electron-ion collision frequency, q is the safety factor, and K(q) and L(Φ_{0}) are functions of q and Φ_{0} respectively. The results are also applicable to an asymmetric potential bump created externally to enhance the inward pinch flux of high energy, collisionless particles.

**Thermonuclear instability of global-type shear Alfven modes **

J.W. Van Dam, G.Y. Fu, C.Z. Cheng

**Abstract**

The effects of thermonuclear alpha particles on the stability of global-type shear Alfven waves in toroidal geometry in an ignition tokamak experiment are described. The presence of finite toroidicity can lead to stabilization of the so-called global shear Alfven eigenmode. However, toroidicity induces a new global shear Alfven eigenmode, which can be strongly destabilized via transit resonance with alpha particles. In the proposed International Thermonuclear Experimental Reactor, due to its large size and low density, this latter mode is found to be benign.

**Nonlinear excitation of magnetic undular instability by convective motion **

M. Kaisig, T. Tajima, K. Shibata, S. Nozawa, R. Matsumoto

**Abstract**

The influence of convective motions on the stabilty of a Parker stable magnetic flux sheet is numerically investigated. The general characteristics of the Parker instability with convective motions in the nonlinear stage are studied. The possibility that a destabilization of stable flux can be realized by either horizontal photospheric shearing motions and/or by vertical convective flows is examined.

**On the ExB velocity correlation function and diffusion coefficient in the two wave system **

D.E. Kim, E.G. Heo, D.I. Choi, W. Horton

**Abstract**

The scaling of the diffusion coefficient with the amplitude of the fluctuation is studied in conjunction with the **E x B** velocity correlation function. For high fluctuation amplitude the velocity correlation function is split into two parts. One part is the correlation for the integrable case which does not contribute to the net transport. The other part is the correlation which results from the stochasticity. The diffusion coefficient obtained from the integrals of the second part of correlation, scales as ∅^{0} or ∅^{-1} depending on the cases considered.

**Point vortex description of drift wave vortices: dynamics and transport **

M. Kono, W. Horton

**Abstract**

Point-vortex description for drift wave vortices is formulated based on the Hasegawa-Mima equation to study elementary processes for the interactions of vortices as well as statistical properties like vortex diffusion. Dynamical properties of drift wave vortices known by numerical experiments are recovered. Furthermore a vortex diffusion model discussed by Horton based on numerical simulations is shown to be analytically obtained. A variety of phenomena arising from the short-range nature of the interaction force of point vortices are suggested.

**The neoclassical transport of toroidal momentum in tokamaks **

A.A. Ware

**Abstract**

Allowing for the presence of an expected moderate concentration (∼10%) of low-energy ions in excess of the Maxwellian distribution causes a dramatic change in the neoclassical prediction for momentum transport, leading to an agreement in magnitude with experiment.

**Magnetohydrodynamic modes driven by anomalous electron viscosity and their role in fast sawtooth crashes **

A.Y. Aydemir

**Abstract**

Dispersion relations are derived for both small- and large-Δ' modes (*m* ≥ 2 and *m*=1 modes, respectively) driven by anomalous electron viscosity. Under the assumption that the anomalous kinematic electron viscosity is comparable to the anomalous electron thermal diffusivity, it is found that the viscous mode typically has a higher growth rate than the corresponding resistive mode. Computational results in cylindrical and toroidal geometries are compared with theory and some nonlinear results for viscous *m*=1 modes in both circular and D-shaped boundaries are presented and their possible role in fast sawtooth crashes is discussed.